| L(s) = 1 | + (1.22 − 0.707i)2-s + (−2.54 + 1.58i)3-s + (0.999 − 1.73i)4-s + (−0.790 + 0.456i)5-s + (−1.99 + 3.74i)6-s − 2.82i·8-s + (3.98 − 8.07i)9-s + (−0.645 + 1.11i)10-s + (12.6 + 7.27i)11-s + (0.196 + 5.99i)12-s − 0.583·13-s + (1.29 − 2.41i)15-s + (−2.00 − 3.46i)16-s + (18.6 + 10.7i)17-s + (−0.832 − 12.7i)18-s + (8 + 13.8i)19-s + ⋯ |
| L(s) = 1 | + (0.612 − 0.353i)2-s + (−0.849 + 0.528i)3-s + (0.249 − 0.433i)4-s + (−0.158 + 0.0913i)5-s + (−0.333 + 0.623i)6-s − 0.353i·8-s + (0.442 − 0.896i)9-s + (−0.0645 + 0.111i)10-s + (1.14 + 0.661i)11-s + (0.0163 + 0.499i)12-s − 0.0448·13-s + (0.0861 − 0.161i)15-s + (−0.125 − 0.216i)16-s + (1.09 + 0.633i)17-s + (−0.0462 − 0.705i)18-s + (0.421 + 0.729i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.158i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.987 - 0.158i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.84219 + 0.147196i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.84219 + 0.147196i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.22 + 0.707i)T \) |
| 3 | \( 1 + (2.54 - 1.58i)T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + (0.790 - 0.456i)T + (12.5 - 21.6i)T^{2} \) |
| 11 | \( 1 + (-12.6 - 7.27i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + 0.583T + 169T^{2} \) |
| 17 | \( 1 + (-18.6 - 10.7i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-8 - 13.8i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-33.6 + 19.4i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 - 35.7iT - 841T^{2} \) |
| 31 | \( 1 + (-29.2 + 50.6i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (10 + 17.3i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + 8.75iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 11.7T + 1.84e3T^{2} \) |
| 47 | \( 1 + (7.34 - 4.24i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-44.0 - 25.4i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (50.4 + 29.1i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (19.4 + 33.7i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (35.2 - 61.1i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 17.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (36.1 - 62.6i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (10.4 + 18.1i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + 145. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (-46.4 + 26.8i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + 111.T + 9.40e3T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.72499536404754190669141502135, −10.76736494766107723989101470411, −9.951062049668659484359450701756, −9.084830193241959413841386389050, −7.40807742795507533192865519053, −6.39024129133424832529010891334, −5.43103000471069557305260455025, −4.34200518376308087807847974206, −3.40722436030286571332777434035, −1.29876680989642741932258011576,
1.07252272802622493402203191793, 3.12411329200142673020708730163, 4.59916203856295548703975743354, 5.55639658320135266030016102767, 6.54906648538572110643770354125, 7.32099232990773720452710761847, 8.427437567451760563833871959539, 9.699351826393769452697355691596, 10.98291203622042405053199590497, 11.84644593976395464333918553765
